Poisson Distribution 95 Confidence Interval Calculator

The Poisson distribution is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval. This calculator computes the 95% confidence interval for a Poisson distribution based on sample data.

What is Poisson Distribution?

The Poisson distribution is commonly used to model the number of events occurring within a fixed interval of time or space. It's characterized by a single parameter λ (lambda), which represents the average rate of events.

Key properties of the Poisson distribution:

Discrete distribution (counts of events)
Parameter λ > 0
Mean = Variance = λ
Applicable when events occur independently at a constant average rate

Common applications include:

Number of phone calls received per hour
Number of accidents at an intersection
Number of emails received per day

What is a 95% Confidence Interval?

A 95% confidence interval is a range of values that is likely to contain the true population parameter with 95% probability. For the Poisson distribution, this interval estimates the true event rate λ.

Key points about confidence intervals:

Not a probability statement about λ
95% refers to the method's reliability, not the probability that λ falls in the interval
Wider intervals provide more confidence but less precision
Narrower intervals provide more precision but less confidence

For large sample sizes (n ≥ 20), the normal approximation can be used. For smaller samples, exact methods are preferred.

How to Calculate Poisson Confidence Interval

The calculation involves these steps:

Collect sample data (count of events)
Calculate the sample mean (λ̂ = x/n)
Determine the critical value from the normal distribution
Calculate the standard error
Compute the confidence interval bounds

Formula:

Lower bound = λ̂ - z*(√λ̂/n)

Upper bound = λ̂ + z*(√λ̂/n)

Where z is the z-score for 95% confidence (approximately 1.96)

The calculator uses this exact method to provide precise results.

Worked Example

Suppose you observe 15 events in 100 trials:

Sample mean (λ̂) = 15/100 = 0.15
Standard error = √(0.15/100) ≈ 0.0387
Z-score for 95% confidence ≈ 1.96
Margin of error = 1.96 * 0.0387 ≈ 0.0756

The 95% confidence interval would be:

0.15 - 0.0756 ≈ 0.0744 to 0.15 + 0.0756 ≈ 0.2256

This means we are 95% confident that the true event rate λ is between approximately 0.0744 and 0.2256.

FAQ

What is the difference between confidence interval and prediction interval?

A confidence interval estimates the true population parameter (λ), while a prediction interval estimates the range for future observations. Confidence intervals are typically narrower than prediction intervals.

When should I use a Poisson distribution instead of a normal distribution?

Use Poisson when modeling count data with a constant average rate and independent events. Use normal when dealing with continuous data or when the sample size is large enough for the Central Limit Theorem to apply.

How does sample size affect the confidence interval width?

Larger sample sizes produce narrower confidence intervals because the standard error decreases with increasing sample size. The relationship is approximately inverse square root: width ∝ 1/√n.